9401538 Boyle Boyle will work on problems involving automorphisms of a shift of finite type, good maps betwen Markov processes, expansive Z^n actions, finite presentations of dynamical systems, topological orbit equivalence and ordered cohomology, inverse problems for nonnegative matrices, and general symbolic dynamics. As specific examples in the last three categories, he hopes with collaborators and students to --characterize the ordered groups which arise as the ordered first Cech cohomology group of a homeomorphism of a zero dimensional dynamical system --characterize the nonzero spectra of integral nonnonegative matrices by solving an associated factorization problem in the ring of formal power series with integral coefficients --extend the definition of residual entropy to smooth dynamical systems and determine whether in that case it is a nontrivial invariant (as it can be in the zero dimensional case). Among the topics proposed by Rudolph for study in measurable dynamics are restricted orbit equivalence and characteristic factors for generalized ergodic theorems. The notion of restricted orbit equivalence is intended as a method to investigate the nature of a dynamical system by considering the perturbations of its orbit structure. The choice of restriction one places on the kind of perturbation allowed controls the type of structure observed. -- In investigating the convergence of various kinds of generalized ergodic theorems, one searches for natural splittings into parts converging to zero, and parts with nontrivial convergence, but very limiting, usually algebraic structure. This latter "characteristic factor" of the averaging method not only controls the convergence, but also gives insight into the dynamics of the underlying system. This is pure mathematical research in the general areas of dynamical systems and matrix theory. The dynamical parts largely involve theoretical problems in symbolic dynamics; this abstract s ubject is involved in practical problems of encoding data in magnetic media, and it has a finite aspect which has lent itself to applicability. (Abstract classficiation schemes gave rise to an algorithm which is now part of an IBM product.) The matrix research is a continuation of a successful, novel application of symbolic dynamics techniques and viewpoint to some old, basic and difficult problems in another subject, the theory of nonnegative matrices. The underlying philosophy of measurable dynamics is to use observations of the behavior of a system over time to gain insight into the structure of the system. In this one can loosen the notion of "time" to include the spatial dimensions of a crystal or the linear dimension of a DNA molecule. Such simple changes of perspective can lead to deep and fruitful insights into how the structure of the system is encoded in its temporal behavior. A central theme of the work is to use such simple real-world models to obtain deeper understanding of dynamical systems. ***
